Kitchen > PrivateWebHome > WebLog > ExtendedProblem6 (r1.60)
13 Dec 2005 - 22:37

## what kind of problem is this?

What is the digit in the hundreds places of the sum of the following addition problem:

7 + 77 + 777 + 7777 + ... + 77777777777777777777

(The final number has 20 7s)

Thanks—

extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal

Back to main page.

After entering a comment, users can login anonymously as KtmGuest (password: guest) when prompted.
Please consider registering as a regular user.
Look here for syntax help.

This is a place-value problem. (I would also term it an algebra problem -- but possibly not in the way that the general public thinks of algebra.)

By what we know about place value, we can solve this problem by disregarding everything to the left of the hundreds place in all of the numbers.

Thus, the sum in question will have the same digit in the hundreds place as would: 7 + 77 + (18)(777).

Or, we can imagine adding in columns, like the standard addition algorithm.

Our right-most colum will yield 20*7. Adding this to the values in the tens place, that result would be added to 19*70. And working with the hundreds, we add on 18*700. If I haven't screwed up, this should be the same number that I defined before.

I don't have a calculator handy, so evaluation is left as an exercise to the reader. :)

-- RudbeckiaHirta - 13 Dec 2005

THANK YOU

-- CatherineJohnson - 13 Dec 2005

I looked up the FERPA stuff today.

Thank you2

-- CatherineJohnson - 13 Dec 2005

Rudbeckia beat me to it. I'll lay out my approach in case it helps too.

I thought of it as follows:

```                    7
+                  77
+                 777
+                 ...
+77777777777777777777
---------------------
```
When I laid it out mentally like that, and thought about what I do when doing multi-digit multiplication. So, start by summing the 1s column. There are 20 7s there. 20 7s = 140.

In the 10s column there are 19 7s. 19 7s = 133. And we should put this over one place.

In the 100s column there are 18 7s. We should put this over two places.

```                    7
+                  77
+                 777
+                 ...
+77777777777777777777
---------------------
140
1330
12600
-------
070
```

To me it's a critical thinking question and a good one for an accelerated class for gifted kids.

-- TracyW - 13 Dec 2005

(20X7) + (19X70) + (18X700) = 140 + 1330 + 12600

1 + 3 + 6 = 10

-- KtmGuest - 13 Dec 2005

Christopher, of course, insisted on writing EVERY 7 out....

-- CatherineJohnson - 13 Dec 2005

now we have sobbing and crying

-- CatherineJohnson - 13 Dec 2005

thank god Ed is dealing with it

-- CatherineJohnson - 13 Dec 2005

And the sum is: 86419753086419700000

-- SmartestTractor - 13 Dec 2005

86419753086419700000

good lord

-- CatherineJohnson - 13 Dec 2005

ok, so Ed and I just went through all his tests, etc.....

basically, our conclusion is that he's being de-instructed

things he knew going into the course are disappearing

fast

-- CatherineJohnson - 13 Dec 2005

I realize that means he didn't know these things to mastery

so what

if he hadn't gone to middle school he would know them to mastery by now, because I would have been teaching them to mastery

-- CatherineJohnson - 13 Dec 2005

we were working our way through ALL the fraction units in PRIMARY MATHEMATICS

that's done

no time

-- CatherineJohnson - 13 Dec 2005

my last hope at this point is KUMON, which can be done on the side

-- CatherineJohnson - 13 Dec 2005

KUMON, summer re-teaching, and I'm going to figure out SOME way to distribute practice of ROTE knowledge

-- CatherineJohnson - 13 Dec 2005

how is this algebra, by the way?

(That reminds me, I want to ask everyone exactly how I should be defining algebra....)

I didn't do problems like this in algebra.

-- CatherineJohnson - 13 Dec 2005

When I laid it out mentally like that, and thought about what I do when doing multi-digit multiplication. So, start by summing the 1s column. There are 20 7s there. 20 7s = 140.

oh, that's cool

Christopher wrote every single number (thank God we had graph paper), then crossed out all the extra digits, then did this:

18 x 777

+

7

+

77

-- CatherineJohnson - 13 Dec 2005

one mom of an incredibly bright little guy in K-3 says her child is losing the math he knew

TRAILBLAZERS is confusing him terribly

that's exactly what we see happening here

the math he knew to some degree is getting jumbled

-- CatherineJohnson - 13 Dec 2005

Christopher really bursts into tears if he can't do a problem?

And Smartest Tractor - I get a 7 in the tens place. Are you sure your answer is right and your computer isn't rounding wrongly?

-- TracyW - 13 Dec 2005

I'm a bit puzzled. To me it seems that the easiest way to do this is to frame the problem as follows:

(18 x 777) + 77 + 7 = ?

I wrote a trivial C program to calculate this, and the answer is 14,070. So unless I've goofed, Tracy's right.

Here's the code:

#include

main() { printf("7 + 77 + (18 * 777) is %d\n", 7 + 77 + (18 * 777)); }

And the output is 14070.

NZ 1, rest of the world 0 :-)

-- VerghisKoshi - 13 Dec 2005

Grr, screwed up the format, should be:

#include

main() { printf("7 + 77 + (18 * 777) is %d\n", 7 + 77 + (18 * 777)); }

-- VerghisKoshi - 13 Dec 2005

I surrender!

-- VerghisKoshi - 13 Dec 2005

Christopher really bursts into tears if he can't do a problem?

Yes.

We are now in a situation where school is so aversive to him that he's crying at night, at home.

Last night, after Ed read to him, Christopher told Ed that he's stupid. He's been telling me the same thing.

He's no longer sitting with his friends at lunch, because they're telling him he's stupid.

Yesterday, one of his oldest friends told him, "Mrs. Roth's a good teacher, you're just stupid."

He's being teased and taunted now by his friends, who have been set on him by Mrs. Roth.

The same thing happened to a child I know in another school. The teacher turned an entire class against the child. That's how bullying works; the bully attracts a group of 'fellow travelers' who've seen what happened to the victim, and don't want to be victims themselves.

The mom I know pulled her kid out, period.

She said to the principal, 'She's not coming back. I look forward to speaking to the truant officer.'

Needless to say, the truant officer didn't show up.

-- CatherineJohnson - 13 Dec 2005

Ed hit the tipping point last night, when Christopher told him he felt stupid, and his friends are calling him stupid.

This is ED, who is actively opposed to vouchers.

He said this morning that when we moved here he couldn't imagine why anyone would send his child to a private school.

Then he said that if we had the money he'd pull Christopher today.

-- CatherineJohnson - 13 Dec 2005

I'd say we're pretty close to homeschooling.

Ed is opposed, and now that I'm actually facing the possibility of trying to homeschool and earn enough money to pay property taxes to support the schools, I'm experiencing pause. (Plus I'd also have to figure out how to teach science and social studies; I'd have even more work to do than I do now reteaching English & math.)

Spanish, I can handle. (He takes Spanish.)

Nevertheless, this can't go on. We were dealing with the poor quality education. I can reteach his material, and now that Ed's fully on board, as opposed to indulgently on board, there's no question Christopher will learn it.

But to reteach everything, and try to undo emotional damage every night—that's not going to work.

I think if we can get him out of Mrs. Roth's class (he has her for two hours a day) he'll be fine.

I tracked the middle school as closely as I could last year.

The kids were crying about math quite a lot. (And I mean a lot. I didn't hear a single tale of a child not crying.)

But they didn't have two hours with a grown-up bully every day as well.

-- CatherineJohnson - 13 Dec 2005

This is ED, who is actively opposed to vouchers.

What's the old joke. A conservative is just a liberal who's been mugged. In this case, educationally mugged.

-- KDeRosa - 13 Dec 2005

What kind of problem is this?

My guess is it's some "thinking outside the box" problem. Here's the problem - the box hasn't been filled yet.

Catherine, I was sharing your site with another homeschooling mom last night. She was shocked and saddened by what you and your child are going through.

You and your child are suffering trying to follow the school's curriculum/schedule that has little regard for what you child is actually mastering. The afterschooling must be difficult for a child who has to be in school for 7 hours a day and then come home to more mental demands. This can't be good for your relationship in the long run.

Please, please, please keep an open mind about homeschooling. Get together with some homeschoolers and see what it's all about. If you choose to do it, remember it's not a forever thing. Give yourself a trial period. Help your child find his love of learning again. I know your husband has doubts, but maybe he will agree to a trial period.

-- NicksMama - 13 Dec 2005

Ed's a historian, so wouldn't it be logical if he took at least social studies? It would mean one less subject that you would be teaching, and would give Christopher a substansial break between doing maths, English, science, etc.

Incidentally, from what I've read about science, it's near impossible to teach real science without calculus already - most of what they doing in school before students do calculus is just junk science. This means that science wouldn't actually be too difficult to handle. Buy a copy of the school's textbook (or a better one) and work your way through it slowly, and do a few science experiments. Speak to the science teacher (I think that's the one who is doing well by her class) and find out exactly what they cover and what books she would actually recommend - the school chosen textbook (as you've already realised in relation to maths) is likely to be... (restraining swearwords) less than useful.

Homeschooling is almost certainly the way to go. If nothing else, the willingness to pull your kid out of the public school will make the admin and teachers really take notice. My mum got the run-around right up until she announced she was taking my brother out of the public school, and then they couldn't do enough for her.

-- SamanthaRawson - 13 Dec 2005

A conservative is just a liberal who's been mugged.

YOU SAID IT!

NOT ME!

Impulse control prevails in these parts!

(I'm serious. I came THIS close to writing, 'Ed has been mugged by reality.')

-- CatherineJohnson - 13 Dec 2005

"This is ED, who is actively opposed to vouchers."

"He said this morning that when we moved here he couldn't imagine why anyone would send his child to a private school."

"What's the old joke. A conservative is just a liberal who's been mugged. In this case, educationally mugged."

This is my story of educational mugging. I would rather call it an enlightenment.

I grew up in public schools in Connecticut. They were great. (not perfect) I have always been a supporter of public schools and have never complained about the taxes. (I still don't.) I never liked the idea of vouchers, especially partial vouchers.

Things have changed.

When I was growing up, nobody ever dreamed of sending their kids to an independent K-8 school. (except for some who sent their kids to Catholic schools) This changed a little in high school because some parents liked the idea of academies, but who would ever pay for private K-8 schooling?!?

Things have changed.

My son is in a private school. (not perfect) There is a boom in enrollment in private K-8 schools in spite of \$10,000 +++ tuition fees. There is a big demand for charter schools slots even though some of these schools are still pretty fuzzy. (parents want to try something ... anything else) Twenty-five percent of the parents in our town send their kids to other schools. Many of these parents grew up in public schools. The main problems? Low expectations and poor curricula. Parents see that their kids are not getting the same education they had when they were growing up.

Public schools can only dismiss these people as elitist for so long.

When my son was in the public schools in first grade, I was a member of the School Improvement Team and was actively involved with the school. I wasn't happy that they were using MathLand, but I tried to be an involved partner for change. I soon realized that the school and teachers really don't want any help in the areas of curriculum, teaching methods, and basic assumptions.

My son was learning very little in their full-inclusion, low expectation environment so I worked with the teacher to get my son more appropriate work. This resulted in a few advanced reading books for the individual reading time in class and some extra homework. I worked with him on this extra homework, but his teacher didn't look at it and didn't test him on his reading at school. All of their focus was on the lowest level students. This is fine up to a point, but when I saw that they were still trying to get some kids in third grade to master their adds and subtracts to 20, I knew there was a big problem.

One parent told me that Kindergarten was great, but it was all downhill from there.

What bothered me the most was a sense that the the school almost didn't like smart kids. First, they would never call them smart kids. Then, in their effort to treat all kids the same, they seemed to ignore the needs of the kids who could be accelerated. For philosophical reasons, they do not accelerate or do pull-out, so perhaps they have to pretend that the more able kids are just not that smart. I heard one of the principals say that despite noticable differences in kids in Kindergarten, these differences go away by fourth grade. (probably because the more able kids are not allowed to get ahead) She is also the one who said that they feed kids all sorts of learning in the early grades, but don't expect to get anything back (from the kids) until fourth grade. This meant that the school sets minimal expectations of knowledge and skills until fourth grade. The only tracking that is done is giving a few 8th graders a little bit of algebra to supplement CMP math.

In the lower grades, they had art shows where the best works got awards. This would never happen for subjects like math or geography (superficial knowledge!). I remember one meeting where the teachers were talking about promoting the school to the public. This is a form of PR that would make parents and taxpayers feel good. Basically, they wanted to use the students to promote their agenda. The idea did not come up because they wanted to give individual students recognition. They were stuck. How could they use great individual works for PR without emphasizing that some kids were smarter (ouch) than others. Now, they seem to be satisfied with trumpeting their "High Performing" rating on the state's trivial test and assuring parents that they are working on Differentiated Learning.

Parents have told me that the best education is a private K-8 school followed by the public high school.

Our K-8 schools do not want any outside influence or interference in educational philosophy, basic assumptions, teaching methods, and curricula, even from the high school. They set low and fuzzy expectations in a full-inclusion environment and pit the special needs parents against the parents of kids able and willing to do more. Parents don't want to get trashed. They send their kids to private schools and wash their hands of the problems.

There is now talk of "tuitioning-in" paying IEP students from other communities because of our special needs programs. This is in spite of an acknowledged problem with an "academic ceiling", full-inclusion, no pull-out, and twenty-five percent of the kids being driven away to other schools. On top of all of this, our schools want to prevent kids from going to charter schools because of their high performing status.

Needless to say, I, a former staunch supporter of public schools, am now a big supporter of full vouchers and unfettered charter schools.

-- SteveH - 13 Dec 2005

FWIW, I'm inclined to agree with Samantha Rawson that the science isn't very serious. Dr. John Hubisz's website contains reviews of many science books, and the reviews look reasonable to me.

A couple of datapoints: He panned Prentice-Hall's Science Explorer series, which my son's school uses. I looked at it found a number of problems: for example, after using pounds for mass, they suddenly claim that force is measured in newtons. They claim that a complete revolution of the Earth around the Sun "is a year" - not that "it takes a year".

Also, I liked Integrated Science by J M LeBel?, which he recommended.

There's also the Calvert School, which will sell you a complete home-schooling kit. They seem to have an excellent reputation, and the grammar that I looked at was extremely good.

BTW, it's amazing that we could see what's in your mind ("Ed has been mugged by reality") even though, as you pointed out, you didn't write it :-)

-- VerghisKoshi - 13 Dec 2005

And Smartest Tractor - I get a 7 in the tens place. Are you sure your answer is right and your computer isn't rounding wrongly?

Ooops Since I am feeling rather un-smart, I will firmly blame Excel for my error.

I worked through the problem this morning with my gang while we waited for the assembly.

-- SmartestTractor - 13 Dec 2005

SteveH?,

Couldn't agree with you more.

We started out as sort of liberal public school supporters. Soon after our son was born we started looking into public schools in detail, and what we found horrified us.

So our kids are now both in a private school (not perfect, but better than the alternative). It burns me to pay taxes on top of the tuition, but I suppose I'm keeping some of these public school unemployables off the street.

But I have to admit that their self-righteousness and constant whining that they need more money, together with their inability to do anything useful gets on my nerves. There are exceptions, but not many.

Getting off my soapbox now...

BTW, have you seen the exchanges with Mike in Texas? Sounds like you have.

-- VerghisKoshi - 13 Dec 2005

"So our kids are now both in a private school (not perfect, but better than the alternative)."

Not perfect, but expensively better. For my wife and I, it came down to whether we could keep our son in public school and make it work ... somehow, even with help after school. The answer was no, not even close.

"BTW, have you seen the exchanges with Mike in Texas? Sounds like you have."

I have been part of the exhchages. He seems unable to accept any reality other that what he sees through his own politically-filtered glasses.

-- SteveH - 13 Dec 2005

"We started out as sort of liberal public school supporters. Soon after our son was born we started looking into public schools in detail, and what we found horrified us."

After my son was born, I remember thinking about how my public school education (especially in math) could have been improved. Then I read a glowing article about TERC in the paper and someone told me that our school uses MathLand.

-- SteveH - 13 Dec 2005

whoa!

I didn't see these stories!

-- CatherineJohnson - 13 Dec 2005

My guess is it's some "thinking outside the box" problem. Here's the problem - the box hasn't been filled yet.

OK, I have to GET THIS ON THE TO-DO LIST.

This is great for Wit & Wisdom.

That's exactly it.

-- CatherineJohnson - 13 Dec 2005

Homeschooling is almost certainly the way to go. If nothing else, the willingness to pull your kid out of the public school will make the admin and teachers really take notice. My mum got the run-around right up until she announced she was taking my brother out of the public school, and then they couldn't do enough for her.

Tell me her story again (unless I'm intruding).

She pulled your brothers out and put them in a private school, right?

-- CatherineJohnson - 13 Dec 2005

We're not going to get huge amounts of run-around on this.

That will be his opening position.

But I suspect we'll move from there pretty quickly.

-- CatherineJohnson - 13 Dec 2005

I wasn't happy that they were using MathLand, but I tried to be an involved partner for change.

famous last words

-- CatherineJohnson - 13 Dec 2005

You and your child are suffering trying to follow the school's curriculum/schedule that has little regard for what you child is actually mastering.

Yes, exactly.

The afterschooling must be difficult for a child who has to be in school for 7 hours a day and then come home to more mental demands.

ditto

This can't be good for your relationship in the long run.

We may have this one figured out now. (I say 'may.')

Ed is now completely on board; We Are As One.

I don't have a lot of insight into the 'psychoanalysis' of families (though I think I will at some point, after the kids are grown and I can reflect back on all the mistakes I made).

However, I've been thinking A LOT about the horrors of public schooling combined with fatherless boys.

I don't think it's great for the mother of an 11-year old boy to be hammering content into him each and every night after school.

And given the way our school is set up, hammering is it. It's my only tool. (I have other stories from other moms of boys this age I'll write up VERY anonymously as I go along. It's the same story everywhere.)

Carolyn and I were talking about how both of our sons are obsessed with their dads right now. Obsessed. Christopher worships his dad; he also worships Christian, whom I've mentioned before.

At this point, Ed needs to be in a leadership role on the afterschooling front.

I think that's what it comes down to.

The problem has been that he hasn't really recognized the need for afterschooling.....he thinks Christopher is super-smart, will get the material in time, he was the same way when he was in 6th grade, etc. All that stuff.

That phase has now concluded.

I am now saying things like, 'Do I have to talk to your father?'

This is hilarious, given my age, past & future politics, Scots-Irish nature, etc. I mean, come on. I grew up on a farm in central Illinois. I spent a lot of time hearing adults say things like, STOP CRYING OR I'LL GIVE YOU SOMETHING TO CRY ABOUT.

My own mother never, ever, not once, said, 'Do I have to speak to your father.'

I started saying it for the first time this week.

Christopher's reaction is hilarious.

Throws him completely off his feed.

He'll say, uncertainly, 'I don't care.'

Then he'll just look at me, like he's trying to figure out what could happen if I Speak To His Father.

for the record: My own parents never said STOP CRYING OR I'LL GIVE YOU SOMETHING TO CRY ABOUT.

But I knew plenty who did.

-- CatherineJohnson - 13 Dec 2005

A couple of nights ago Ed said, 'Just call me Ward.'

-- CatherineJohnson - 13 Dec 2005

I heard one of the principals say that despite noticable differences in kids in Kindergarten, these differences go away by fourth grade.

That gives me the chills.

-- CatherineJohnson - 13 Dec 2005

assuring parents that they are working on Differentiated Learning

Our district is heavy into Differentiated Instruction at this point

We've hired a specialist in Differentiated Math Instruction.

-- CatherineJohnson - 13 Dec 2005

There is now talk of "tuitioning-in" paying IEP students from other communities because of our special needs programs.

wow

I've never heard that expression before

we've done some of that

-- CatherineJohnson - 13 Dec 2005

Needless to say, I, a former staunch supporter of public schools, am now a big supporter of full vouchers and unfettered charter schools.

yup

-- CatherineJohnson - 13 Dec 2005

Verghis

BTW, it's amazing that we could see what's in your mind ("Ed has been mugged by reality") even though, as you pointed out, you didn't write it :-)

That's funny!

You're right; I didn't think of that.

I was forcing myself not to write the words 'mugged by reality.'

He's been mugged alright (though he may be able to resist becoming a voucher advocate. He's pretty stubborn.)

He said today, unequivocally, that if we had the money Christopher would be in private school today.

That's a huge change for him. Huge.

-- CatherineJohnson - 13 Dec 2005

assuring parents that they are working on Differentiated Learning

"Our district is heavy into Differentiated Instruction at this point"

They used to call it Differentiated Instruction, but that seemed to imply some sort of responsibility on the teacher's part. Just like they used to call it an "academic ceiling", which implies a problem with the schools. Now they call it a "performance ceiling", which puts the problem and onus on the student.

-- SteveH - 13 Dec 2005

"He said today, unequivocally, that if we had the money Christopher would be in private school today."

Exactly. Absent the property-tax burden, many people could choose a school that they like. And the taxes are due as long as you live. It would be cheaper in many cases to just pay tuition while your kid's in school.

Or if that's too radical, then full vouchers.

-- VerghisKoshi - 13 Dec 2005

Steve

They used to call it Differentiated Instruction, but that seemed to imply some sort of responsibility on the teacher's part. Just like they used to call it an "academic ceiling", which implies a problem with the schools. Now they call it a "performance ceiling", which puts the problem and onus on the student.

Straight to Wit & Wisdom.

But first I have to go bang my head against the wall.

-- CatherineJohnson - 13 Dec 2005

WebLogForm
Title: what kind of problem is this?
TopicType: WebLog
SubjectArea: MiddleSchoolMath
LogDate: 200512122012